Use the Gauss-Jordan method to solve each system of equations. Fo...

Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. 3x + 2y = -9 2x - 5y = -6

Accepted Answer

Hello, everyone. We are asked to solve the system of equations using the Gauss Jordan method. The system of equations we are given is comprised of two equations. The first being five X plus three Y equals 35. And the second being seven X minus four, Y equals 49 we have four answer choices, all of which with slightly varying values for X and Y. First thing to recall is that in the Gauss Jordan method, we are going to create an augmented matrix using the coefficients of the variables. And what the equation equals the first row will come from the first equation. So we will have 53 and the second row comes from the second equation seven negative 4, 49 closing the matrix. And because it's an augmented matrix, there is a vertical line between the 2nd and 3rd elements of the rows. Recall that in the Gauss Jordan method, we are allowed to switch row locations multiply a row by a value or add a multiple of a row to the other row. So the first thing I'm going to do is I want to get the first element of the first row to be won. We do this by multiplying row one by 1/5. So when I do so I am left with one 3/5 7 as my first row, my second row did not change in this step. So it is still seven negative 4 49. My next step has a couple moving pieces. In this step. I'm gonna multiply the first row by negative seven and add it to the second row so that the first element of the second row becomes zero because I want one element to be zero and one element to be one in each row. So again, here row two is now gonna be formed by the product of negative seven times row one added to what the current value of row two is. So the first row here will not change. And in row two, row two will be row one. So one from the first element multiplied by negative seven, which is negative seven plus the seven that was in row two, which will give us zero next 3/ multiplied by negative seven, which is negative 21 5th added to negative four, which ends up being negative 41 5th and negative seven multiplied by seven from row one is negative 49 added to the 49. That is already in row two gives us zero. So our second row now reads zero, negative 41 5th zero. So we have our zeros in this second row, we want to find a way to make negative 41 5th into a one. So our next step is gonna be in this next row here and I'm going to multiply row two by negative 5 firsts. So negative 5/41. So row two is gonna be multiplied by negative 5/ row one is not changing at this step. So we have 1 3/5 and for row two zero, multiplied by negative 5 40 firsts is going to still be zero, negative five 41st multiplied by negative 41 5th will give us a positive one. And for the last element, zero times anything is zero. So we now have our bottom row reading 010. And that's what we wanted. We wanted to figure out how to have only ones and zeros in the left hand side. So now I need in the first row to make 3/5 into a zero. So I need to think of a way to do that. So that way I can find an X value and A Y value out of this. And just because row two looks how we want it to doesn't mean we can't use it as a help for us. So to get row one, because we want that 3/5 to go away. Row one is going to equal negative 3/ times row two added to row one. When I do this, the 3/5. That's in the second element of the first row should go away. So let's step through it row two. So the first element of row two is a zero. OK, zero times negative 3/5 is still zero plus one is still one second element one from row two multiplied by negative 3/5 is negative 3/5 added to 3/5 is zero, which is what we wanted to happen. And then zero from row two, multiplied by negative 3/5 is zero plus the seven from row one is still seven row two will not change. It is still 010. So this actually is enough information for us to find the answer. If you think back to our equation, the first elements were the coefficients of X. So for row one, we have X is one, Y is zero and equals seven. So this is telling us that one, X equals seven for row two, we'd have zero X S one Y and that equals zero. So this means that Y equals zero. So in the end, we found out that X is seven Y is zero and that's answer choice. A have a nice day.